\(\int (b+2 c x) (d+e x)^4 (a+b x+c x^2) \, dx\) [515]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^6}{6 e^4}-\frac {3 c (2 c d-b e) (d+e x)^7}{7 e^4}+\frac {c^2 (d+e x)^8}{4 e^4} \] Output:

-1/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^5/e^4+1/6*(6*c^2*d^2+b^2*e^2 
-2*c*e*(-a*e+3*b*d))*(e*x+d)^6/e^4-3/7*c*(-b*e+2*c*d)*(e*x+d)^7/e^4+1/4*c^ 
2*(e*x+d)^8/e^4
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=a b d^4 x+\frac {1}{2} d^3 \left (b^2 d+2 a c d+4 a b e\right ) x^2+\frac {1}{3} d^2 \left (3 b c d^2+4 b^2 d e+8 a c d e+6 a b e^2\right ) x^3+\frac {1}{2} d \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^4+\frac {1}{5} e \left (8 c^2 d^3+b e^2 (4 b d+a e)+2 c d e (9 b d+4 a e)\right ) x^5+\frac {1}{6} e^2 \left (12 c^2 d^2+b^2 e^2+2 c e (6 b d+a e)\right ) x^6+\frac {1}{7} c e^3 (8 c d+3 b e) x^7+\frac {1}{4} c^2 e^4 x^8 \] Input:

Integrate[(b + 2*c*x)*(d + e*x)^4*(a + b*x + c*x^2),x]
 

Output:

a*b*d^4*x + (d^3*(b^2*d + 2*a*c*d + 4*a*b*e)*x^2)/2 + (d^2*(3*b*c*d^2 + 4* 
b^2*d*e + 8*a*c*d*e + 6*a*b*e^2)*x^3)/3 + (d*(c^2*d^3 + 6*c*d*e*(b*d + a*e 
) + b*e^2*(3*b*d + 2*a*e))*x^4)/2 + (e*(8*c^2*d^3 + b*e^2*(4*b*d + a*e) + 
2*c*d*e*(9*b*d + 4*a*e))*x^5)/5 + (e^2*(12*c^2*d^2 + b^2*e^2 + 2*c*e*(6*b* 
d + a*e))*x^6)/6 + (c*e^3*(8*c*d + 3*b*e)*x^7)/7 + (c^2*e^4*x^8)/4
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^3}+\frac {(d+e x)^4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {3 c (d+e x)^6 (2 c d-b e)}{e^3}+\frac {2 c^2 (d+e x)^7}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^4}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {3 c (d+e x)^7 (2 c d-b e)}{7 e^4}+\frac {c^2 (d+e x)^8}{4 e^4}\)

Input:

Int[(b + 2*c*x)*(d + e*x)^4*(a + b*x + c*x^2),x]
 

Output:

-1/5*((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5)/e^4 + ((6*c^2*d^2 
 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^6)/(6*e^4) - (3*c*(2*c*d - b*e 
)*(d + e*x)^7)/(7*e^4) + (c^2*(d + e*x)^8)/(4*e^4)
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(241\) vs. \(2(116)=232\).

Time = 0.68 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95

method	result	size
norman	\(\frac {c^{2} e^{4} x^{8}}{4}+\left (\frac {3}{7} c \,e^{4} b +\frac {8}{7} c^{2} d \,e^{3}\right ) x^{7}+\left (\frac {1}{3} e^{4} c a +\frac {1}{6} b^{2} e^{4}+2 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right ) x^{6}+\left (\frac {1}{5} e^{4} a b +\frac {8}{5} d \,e^{3} a c +\frac {4}{5} b^{2} d \,e^{3}+\frac {18}{5} d^{2} e^{2} b c +\frac {8}{5} d^{3} e \,c^{2}\right ) x^{5}+\left (d \,e^{3} a b +3 a c \,d^{2} e^{2}+\frac {3}{2} d^{2} e^{2} b^{2}+3 b c \,d^{3} e +\frac {1}{2} c^{2} d^{4}\right ) x^{4}+\left (2 d^{2} e^{2} a b +\frac {8}{3} a c \,d^{3} e +\frac {4}{3} b^{2} d^{3} e +d^{4} b c \right ) x^{3}+\left (2 d^{3} e a b +a c \,d^{4}+\frac {1}{2} b^{2} d^{4}\right ) x^{2}+a b \,d^{4} x\)	\(242\)
gosper	\(3 x^{4} b c \,d^{3} e +2 x^{6} d^{2} e^{2} c^{2}+\frac {4}{5} x^{5} b^{2} d \,e^{3}+\frac {18}{5} x^{5} d^{2} e^{2} b c +3 x^{4} a c \,d^{2} e^{2}+\frac {4}{3} b^{2} d^{3} e \,x^{3}+\frac {3}{2} x^{4} d^{2} e^{2} b^{2}+\frac {1}{5} x^{5} e^{4} a b +2 a b \,d^{2} e^{2} x^{3}+2 x^{2} d^{3} e a b +x^{4} d \,e^{3} a b +\frac {1}{2} c^{2} d^{4} x^{4}+a c \,d^{4} x^{2}+2 x^{6} d \,e^{3} b c +b c \,d^{4} x^{3}+\frac {8}{5} x^{5} d \,e^{3} a c +\frac {3}{7} x^{7} c \,e^{4} b +\frac {8}{3} x^{3} a c \,d^{3} e +\frac {8}{7} x^{7} c^{2} d \,e^{3}+\frac {1}{3} x^{6} e^{4} c a +\frac {8}{5} x^{5} d^{3} e \,c^{2}+a b \,d^{4} x +\frac {1}{6} b^{2} e^{4} x^{6}+\frac {1}{4} c^{2} e^{4} x^{8}+\frac {1}{2} x^{2} b^{2} d^{4}\)	\(281\)
risch	\(3 x^{4} b c \,d^{3} e +2 x^{6} d^{2} e^{2} c^{2}+\frac {4}{5} x^{5} b^{2} d \,e^{3}+\frac {18}{5} x^{5} d^{2} e^{2} b c +3 x^{4} a c \,d^{2} e^{2}+\frac {4}{3} b^{2} d^{3} e \,x^{3}+\frac {3}{2} x^{4} d^{2} e^{2} b^{2}+\frac {1}{5} x^{5} e^{4} a b +2 a b \,d^{2} e^{2} x^{3}+2 x^{2} d^{3} e a b +x^{4} d \,e^{3} a b +\frac {1}{2} c^{2} d^{4} x^{4}+a c \,d^{4} x^{2}+2 x^{6} d \,e^{3} b c +b c \,d^{4} x^{3}+\frac {8}{5} x^{5} d \,e^{3} a c +\frac {3}{7} x^{7} c \,e^{4} b +\frac {8}{3} x^{3} a c \,d^{3} e +\frac {8}{7} x^{7} c^{2} d \,e^{3}+\frac {1}{3} x^{6} e^{4} c a +\frac {8}{5} x^{5} d^{3} e \,c^{2}+a b \,d^{4} x +\frac {1}{6} b^{2} e^{4} x^{6}+\frac {1}{4} c^{2} e^{4} x^{8}+\frac {1}{2} x^{2} b^{2} d^{4}\)	\(281\)
parallelrisch	\(3 x^{4} b c \,d^{3} e +2 x^{6} d^{2} e^{2} c^{2}+\frac {4}{5} x^{5} b^{2} d \,e^{3}+\frac {18}{5} x^{5} d^{2} e^{2} b c +3 x^{4} a c \,d^{2} e^{2}+\frac {4}{3} b^{2} d^{3} e \,x^{3}+\frac {3}{2} x^{4} d^{2} e^{2} b^{2}+\frac {1}{5} x^{5} e^{4} a b +2 a b \,d^{2} e^{2} x^{3}+2 x^{2} d^{3} e a b +x^{4} d \,e^{3} a b +\frac {1}{2} c^{2} d^{4} x^{4}+a c \,d^{4} x^{2}+2 x^{6} d \,e^{3} b c +b c \,d^{4} x^{3}+\frac {8}{5} x^{5} d \,e^{3} a c +\frac {3}{7} x^{7} c \,e^{4} b +\frac {8}{3} x^{3} a c \,d^{3} e +\frac {8}{7} x^{7} c^{2} d \,e^{3}+\frac {1}{3} x^{6} e^{4} c a +\frac {8}{5} x^{5} d^{3} e \,c^{2}+a b \,d^{4} x +\frac {1}{6} b^{2} e^{4} x^{6}+\frac {1}{4} c^{2} e^{4} x^{8}+\frac {1}{2} x^{2} b^{2} d^{4}\)	\(281\)
orering	\(\frac {x \left (105 c^{2} e^{4} x^{7}+180 b c \,e^{4} x^{6}+480 c^{2} d \,e^{3} x^{6}+140 a c \,e^{4} x^{5}+70 b^{2} e^{4} x^{5}+840 b c d \,e^{3} x^{5}+840 e^{2} c^{2} d^{2} x^{5}+84 a b \,e^{4} x^{4}+672 a c d \,e^{3} x^{4}+336 b^{2} d \,e^{3} x^{4}+1512 b c \,d^{2} e^{2} x^{4}+672 c^{2} d^{3} e \,x^{4}+420 a b d \,e^{3} x^{3}+1260 a c \,d^{2} e^{2} x^{3}+630 b^{2} d^{2} e^{2} x^{3}+1260 b c \,d^{3} e \,x^{3}+210 c^{2} d^{4} x^{3}+840 a b \,d^{2} e^{2} x^{2}+1120 a c \,d^{3} e \,x^{2}+560 b^{2} d^{3} e \,x^{2}+420 b c \,d^{4} x^{2}+840 a b \,d^{3} e x +420 a c \,d^{4} x +210 b^{2} d^{4} x +420 a b \,d^{4}\right )}{420}\)	\(281\)
default	\(\frac {c^{2} e^{4} x^{8}}{4}+\frac {\left (\left (b \,e^{4}+8 c d \,e^{3}\right ) c +2 c \,e^{4} b \right ) x^{7}}{7}+\frac {\left (\left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) c +\left (b \,e^{4}+8 c d \,e^{3}\right ) b +2 e^{4} c a \right ) x^{6}}{6}+\frac {\left (\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) c +\left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) b +\left (b \,e^{4}+8 c d \,e^{3}\right ) a \right ) x^{5}}{5}+\frac {\left (\left (4 b \,d^{3} e +2 c \,d^{4}\right ) c +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) b +\left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) a \right ) x^{4}}{4}+\frac {\left (d^{4} b c +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) b +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a \right ) x^{3}}{3}+\frac {\left (b^{2} d^{4}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) a \right ) x^{2}}{2}+a b \,d^{4} x\)	\(290\)

Input:

int((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/4*c^2*e^4*x^8+(3/7*c*e^4*b+8/7*c^2*d*e^3)*x^7+(1/3*e^4*c*a+1/6*b^2*e^4+2 
*d*e^3*b*c+2*d^2*e^2*c^2)*x^6+(1/5*e^4*a*b+8/5*d*e^3*a*c+4/5*b^2*d*e^3+18/ 
5*d^2*e^2*b*c+8/5*d^3*e*c^2)*x^5+(d*e^3*a*b+3*a*c*d^2*e^2+3/2*d^2*e^2*b^2+ 
3*b*c*d^3*e+1/2*c^2*d^4)*x^4+(2*d^2*e^2*a*b+8/3*a*c*d^3*e+4/3*b^2*d^3*e+d^ 
4*b*c)*x^3+(2*d^3*e*a*b+a*c*d^4+1/2*b^2*d^4)*x^2+a*b*d^4*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {1}{7} \, {\left (8 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{7} + a b d^{4} x + \frac {1}{6} \, {\left (12 \, c^{2} d^{2} e^{2} + 12 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (8 \, c^{2} d^{3} e + 18 \, b c d^{2} e^{2} + a b e^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (c^{2} d^{4} + 6 \, b c d^{3} e + 2 \, a b d e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{4} + 6 \, a b d^{2} e^{2} + 4 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b d^{3} e + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{2} \] Input:

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

1/4*c^2*e^4*x^8 + 1/7*(8*c^2*d*e^3 + 3*b*c*e^4)*x^7 + a*b*d^4*x + 1/6*(12* 
c^2*d^2*e^2 + 12*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^6 + 1/5*(8*c^2*d^3*e + 1 
8*b*c*d^2*e^2 + a*b*e^4 + 4*(b^2 + 2*a*c)*d*e^3)*x^5 + 1/2*(c^2*d^4 + 6*b* 
c*d^3*e + 2*a*b*d*e^3 + 3*(b^2 + 2*a*c)*d^2*e^2)*x^4 + 1/3*(3*b*c*d^4 + 6* 
a*b*d^2*e^2 + 4*(b^2 + 2*a*c)*d^3*e)*x^3 + 1/2*(4*a*b*d^3*e + (b^2 + 2*a*c 
)*d^4)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (114) = 228\).

Time = 0.04 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.25 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=a b d^{4} x + \frac {c^{2} e^{4} x^{8}}{4} + x^{7} \cdot \left (\frac {3 b c e^{4}}{7} + \frac {8 c^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {a c e^{4}}{3} + \frac {b^{2} e^{4}}{6} + 2 b c d e^{3} + 2 c^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {a b e^{4}}{5} + \frac {8 a c d e^{3}}{5} + \frac {4 b^{2} d e^{3}}{5} + \frac {18 b c d^{2} e^{2}}{5} + \frac {8 c^{2} d^{3} e}{5}\right ) + x^{4} \left (a b d e^{3} + 3 a c d^{2} e^{2} + \frac {3 b^{2} d^{2} e^{2}}{2} + 3 b c d^{3} e + \frac {c^{2} d^{4}}{2}\right ) + x^{3} \cdot \left (2 a b d^{2} e^{2} + \frac {8 a c d^{3} e}{3} + \frac {4 b^{2} d^{3} e}{3} + b c d^{4}\right ) + x^{2} \cdot \left (2 a b d^{3} e + a c d^{4} + \frac {b^{2} d^{4}}{2}\right ) \] Input:

integrate((2*c*x+b)*(e*x+d)**4*(c*x**2+b*x+a),x)
 

Output:

a*b*d**4*x + c**2*e**4*x**8/4 + x**7*(3*b*c*e**4/7 + 8*c**2*d*e**3/7) + x* 
*6*(a*c*e**4/3 + b**2*e**4/6 + 2*b*c*d*e**3 + 2*c**2*d**2*e**2) + x**5*(a* 
b*e**4/5 + 8*a*c*d*e**3/5 + 4*b**2*d*e**3/5 + 18*b*c*d**2*e**2/5 + 8*c**2* 
d**3*e/5) + x**4*(a*b*d*e**3 + 3*a*c*d**2*e**2 + 3*b**2*d**2*e**2/2 + 3*b* 
c*d**3*e + c**2*d**4/2) + x**3*(2*a*b*d**2*e**2 + 8*a*c*d**3*e/3 + 4*b**2* 
d**3*e/3 + b*c*d**4) + x**2*(2*a*b*d**3*e + a*c*d**4 + b**2*d**4/2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {1}{7} \, {\left (8 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{7} + a b d^{4} x + \frac {1}{6} \, {\left (12 \, c^{2} d^{2} e^{2} + 12 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (8 \, c^{2} d^{3} e + 18 \, b c d^{2} e^{2} + a b e^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (c^{2} d^{4} + 6 \, b c d^{3} e + 2 \, a b d e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{4} + 6 \, a b d^{2} e^{2} + 4 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b d^{3} e + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{2} \] Input:

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

1/4*c^2*e^4*x^8 + 1/7*(8*c^2*d*e^3 + 3*b*c*e^4)*x^7 + a*b*d^4*x + 1/6*(12* 
c^2*d^2*e^2 + 12*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^6 + 1/5*(8*c^2*d^3*e + 1 
8*b*c*d^2*e^2 + a*b*e^4 + 4*(b^2 + 2*a*c)*d*e^3)*x^5 + 1/2*(c^2*d^4 + 6*b* 
c*d^3*e + 2*a*b*d*e^3 + 3*(b^2 + 2*a*c)*d^2*e^2)*x^4 + 1/3*(3*b*c*d^4 + 6* 
a*b*d^2*e^2 + 4*(b^2 + 2*a*c)*d^3*e)*x^3 + 1/2*(4*a*b*d^3*e + (b^2 + 2*a*c 
)*d^4)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (116) = 232\).

Time = 0.19 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.26 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {8}{7} \, c^{2} d e^{3} x^{7} + \frac {3}{7} \, b c e^{4} x^{7} + 2 \, c^{2} d^{2} e^{2} x^{6} + 2 \, b c d e^{3} x^{6} + \frac {1}{6} \, b^{2} e^{4} x^{6} + \frac {1}{3} \, a c e^{4} x^{6} + \frac {8}{5} \, c^{2} d^{3} e x^{5} + \frac {18}{5} \, b c d^{2} e^{2} x^{5} + \frac {4}{5} \, b^{2} d e^{3} x^{5} + \frac {8}{5} \, a c d e^{3} x^{5} + \frac {1}{5} \, a b e^{4} x^{5} + \frac {1}{2} \, c^{2} d^{4} x^{4} + 3 \, b c d^{3} e x^{4} + \frac {3}{2} \, b^{2} d^{2} e^{2} x^{4} + 3 \, a c d^{2} e^{2} x^{4} + a b d e^{3} x^{4} + b c d^{4} x^{3} + \frac {4}{3} \, b^{2} d^{3} e x^{3} + \frac {8}{3} \, a c d^{3} e x^{3} + 2 \, a b d^{2} e^{2} x^{3} + \frac {1}{2} \, b^{2} d^{4} x^{2} + a c d^{4} x^{2} + 2 \, a b d^{3} e x^{2} + a b d^{4} x \] Input:

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

1/4*c^2*e^4*x^8 + 8/7*c^2*d*e^3*x^7 + 3/7*b*c*e^4*x^7 + 2*c^2*d^2*e^2*x^6 
+ 2*b*c*d*e^3*x^6 + 1/6*b^2*e^4*x^6 + 1/3*a*c*e^4*x^6 + 8/5*c^2*d^3*e*x^5 
+ 18/5*b*c*d^2*e^2*x^5 + 4/5*b^2*d*e^3*x^5 + 8/5*a*c*d*e^3*x^5 + 1/5*a*b*e 
^4*x^5 + 1/2*c^2*d^4*x^4 + 3*b*c*d^3*e*x^4 + 3/2*b^2*d^2*e^2*x^4 + 3*a*c*d 
^2*e^2*x^4 + a*b*d*e^3*x^4 + b*c*d^4*x^3 + 4/3*b^2*d^3*e*x^3 + 8/3*a*c*d^3 
*e*x^3 + 2*a*b*d^2*e^2*x^3 + 1/2*b^2*d^4*x^2 + a*c*d^4*x^2 + 2*a*b*d^3*e*x 
^2 + a*b*d^4*x
 

Mupad [B] (verification not implemented)

Time = 12.01 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.92 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=x^5\,\left (\frac {4\,b^2\,d\,e^3}{5}+\frac {18\,b\,c\,d^2\,e^2}{5}+\frac {a\,b\,e^4}{5}+\frac {8\,c^2\,d^3\,e}{5}+\frac {8\,a\,c\,d\,e^3}{5}\right )+x^6\,\left (\frac {b^2\,e^4}{6}+2\,b\,c\,d\,e^3+2\,c^2\,d^2\,e^2+\frac {a\,c\,e^4}{3}\right )+x^4\,\left (\frac {3\,b^2\,d^2\,e^2}{2}+3\,b\,c\,d^3\,e+a\,b\,d\,e^3+\frac {c^2\,d^4}{2}+3\,a\,c\,d^2\,e^2\right )+x^2\,\left (\frac {b^2\,d^4}{2}+2\,a\,e\,b\,d^3+a\,c\,d^4\right )+x^3\,\left (\frac {4\,b^2\,d^3\,e}{3}+c\,b\,d^4+2\,a\,b\,d^2\,e^2+\frac {8\,a\,c\,d^3\,e}{3}\right )+\frac {c^2\,e^4\,x^8}{4}+\frac {c\,e^3\,x^7\,\left (3\,b\,e+8\,c\,d\right )}{7}+a\,b\,d^4\,x \] Input:

int((b + 2*c*x)*(d + e*x)^4*(a + b*x + c*x^2),x)
 

Output:

x^5*((4*b^2*d*e^3)/5 + (8*c^2*d^3*e)/5 + (a*b*e^4)/5 + (8*a*c*d*e^3)/5 + ( 
18*b*c*d^2*e^2)/5) + x^6*((b^2*e^4)/6 + 2*c^2*d^2*e^2 + (a*c*e^4)/3 + 2*b* 
c*d*e^3) + x^4*((c^2*d^4)/2 + (3*b^2*d^2*e^2)/2 + a*b*d*e^3 + 3*b*c*d^3*e 
+ 3*a*c*d^2*e^2) + x^2*((b^2*d^4)/2 + a*c*d^4 + 2*a*b*d^3*e) + x^3*((4*b^2 
*d^3*e)/3 + b*c*d^4 + (8*a*c*d^3*e)/3 + 2*a*b*d^2*e^2) + (c^2*e^4*x^8)/4 + 
 (c*e^3*x^7*(3*b*e + 8*c*d))/7 + a*b*d^4*x
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.26 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {x \left (105 c^{2} e^{4} x^{7}+180 b c \,e^{4} x^{6}+480 c^{2} d \,e^{3} x^{6}+140 a c \,e^{4} x^{5}+70 b^{2} e^{4} x^{5}+840 b c d \,e^{3} x^{5}+840 c^{2} d^{2} e^{2} x^{5}+84 a b \,e^{4} x^{4}+672 a c d \,e^{3} x^{4}+336 b^{2} d \,e^{3} x^{4}+1512 b c \,d^{2} e^{2} x^{4}+672 c^{2} d^{3} e \,x^{4}+420 a b d \,e^{3} x^{3}+1260 a c \,d^{2} e^{2} x^{3}+630 b^{2} d^{2} e^{2} x^{3}+1260 b c \,d^{3} e \,x^{3}+210 c^{2} d^{4} x^{3}+840 a b \,d^{2} e^{2} x^{2}+1120 a c \,d^{3} e \,x^{2}+560 b^{2} d^{3} e \,x^{2}+420 b c \,d^{4} x^{2}+840 a b \,d^{3} e x +420 a c \,d^{4} x +210 b^{2} d^{4} x +420 a b \,d^{4}\right )}{420} \] Input:

int((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a),x)
 

Output:

(x*(420*a*b*d**4 + 840*a*b*d**3*e*x + 840*a*b*d**2*e**2*x**2 + 420*a*b*d*e 
**3*x**3 + 84*a*b*e**4*x**4 + 420*a*c*d**4*x + 1120*a*c*d**3*e*x**2 + 1260 
*a*c*d**2*e**2*x**3 + 672*a*c*d*e**3*x**4 + 140*a*c*e**4*x**5 + 210*b**2*d 
**4*x + 560*b**2*d**3*e*x**2 + 630*b**2*d**2*e**2*x**3 + 336*b**2*d*e**3*x 
**4 + 70*b**2*e**4*x**5 + 420*b*c*d**4*x**2 + 1260*b*c*d**3*e*x**3 + 1512* 
b*c*d**2*e**2*x**4 + 840*b*c*d*e**3*x**5 + 180*b*c*e**4*x**6 + 210*c**2*d* 
*4*x**3 + 672*c**2*d**3*e*x**4 + 840*c**2*d**2*e**2*x**5 + 480*c**2*d*e**3 
*x**6 + 105*c**2*e**4*x**7))/420